We now place this paper in context, with a focus on previous studies of gravitating vortices that we referred to earlier [1]}, [2]}, [3]}, [4]}. It was recognized a long time ago that Einstein gravity in \(2+1\)  dimensions yields a locally flat spacetime outside localized sources, albeit with the topology of a cone [5]}. However, things become interesting when one includes a negative cosmological constant as this leads to the famous BTZ black holes [6]}, [7]}. Later, in a higher-derivative extension of Einstein gravity in \(2+1\)  dimensions called Bergshoeff-Hohm-Townsend (BHT) massive gravity [8]}, black hole solutions in both de Sitter and anti-de Sitter space were found as well as wormhole solutions, kinks, and gravitational solitons [9]}. An analytical study of black holes with spherical scalar hair in AdS\(_3\)  was then later studied [1]}. Closer to our topic of interest, they also constructed black hole vortex solutions with a complex scalar field. These solutions departed from the conventional non-singular vortex in two ways. The scalar field had a singularity at the origin and asymptotically tended towards zero which satisfied the Breitenlohner-Freedman bound [11]} in AdS\(_3\)  but was not the minimum of the potential. In [2]}, how vortices affect the tunneling decay of a false vacuum under Einstein gravity was studied and it was found that compared to Coleman-de Luccia bubbles [13]} the tunneling exponent was less by a factor of a half. Hence vortices are short-lived and become of cosmological interest [2]}. The non-singular vortex under Einstein gravity in an AdS\(_3\)  and Minkowski background was first studied in [3]}. These were not black hole solutions as in [1]}. Non-singular vortex solutions were found numerically for different values of the cosmological constant \(\Lambda \) , VEV \(v\)  and winding number \(n\) . Two expressions for the (ADM) mass of the vortex were obtained: one in terms of the metric and one as an integral overly purely matter fields. The latter showed that the mass was roughly proportional to \(n^2\,v^2\)  (an \(n^2\)  dependence had also been found in [1]}). The mass of the vortex increased as the magnitude of the cosmological constant increased and led to a slightly smaller core for the vortex. Later, work was then extended to include singular vortex solutions besides non-singular ones [4]}. Vortices with conical singularities were obtained in flat backgrounds and BTZ black hole solutions were obtained in curved backgrounds, though it was found that the vortex cannot ultimately hold a black hole at its core [4]}. Our present paper introduces the non-minimal coupling term which is missing in all previous studies of gravitating vortices. As previously pointed out, this term preserves the local \(U(1)\)  gauge invariance of the vortex and is therefore a perfectly natural candidate to add to the action when gravity is present. We already discussed how this term changes the physics significantly, qualitatively.
